1. Field of the Invention
The present invention relates generally to methods and systems for calculating derivative variables and, more particularly, to methods and systems for performing stabilized Monte Carlo simulations suitable for valuating assets and securities in the finance industry, including interest-rate sensitive derivative securities.
2. Description of Related Art
Financial entities expend considerable resources to valuate assets, especially securities and derivatives with uncertain payments linked to interest rate movements, credit, equities, foreign exchange and other factors. Examples of such assets include bond options, options on bond futures, swaptions, lookback options, and nearly the entire gamut of mortgage-backed securities (MBS).
The starting point for valuating a derivative security is typically a stochastic model of an interest rate process referred to as a Term Structure Model and the accompanying cash flows with “risk-adjusted” probabilities. With the risk-adjustment, model asset valuations are the probability weighted average discounted values of the cash flows over all possible yield curve evolutions.
In certain circumstances, the present value calculation can be performed analytically by finding an analytically tractable closed form solution or by numerically calculating a stochastic integral. However, it is infeasible for securities with interest-rate path dependent cash flows, such as most MBS. Accordingly, to valuate these securities or to estimate their change in value, some form of Generalized Monte Carlo (GMC) method is generally used.
Monte Carlo (MC) and GMC methods have widespread application in finance for valuation of complex financial derivatives, initially spurred by the publication of Phelan Boyle, “Options: a Monte Carlo approach,” Journal of Financial Economics, Vol. 4, pp 323-328, (1977). Since then there have been numerous academic research articles on financial application of Monte Carlo methods, many of which are cited in the survey paper authored by Phelan Boyle, Mark Broadie, and Paul Glasserman, titled “Monte Carlo Methods For Security Pricing,” Journal of Economic Dynamics and Control, Vol. 21, pp 1267-1321, (1997). A more recent and comprehensive exposition authored by Peter Jäckel is: “Monte Carlo Methods In Finance,” Wiley (2002).
The literature is centered on the speed and accuracy of individual calculations and techniques for improvement of MC and GMC. Frequently, in the financial industry, the primary focus of the simulations is not the theoretical valuation, per se, but the impact of a parameter perturbation. In the context of valuating securities, such evaluation focuses on, for example, the accuracy of a price prediction on a given day. If both an initial valuation estimate and a parameter adjusted estimate are highly accurate, then the estimated theoretical change will also be highly accurate. Thus, much work has been directed to increasing the accuracy of MC and GMC calculations and techniques.
Standard implementations of the GMC procedures may have discontinuous variation in the sample paths of interest rates in response to parameter perturbations such as slight changes in current interest rates or prices of interest rate derivatives. Therefore, minor changes can cause the estimation errors to vary significantly. One common example of this is day-to-day variations in the market environment with correspondingly small changes in the model parameters. Although this should lead to small changes in the derived variables and small changes to the asset valuations, small changes may not be the result due to instability of the GMC estimation.
More specifically, in most GMC implementations, some of the derived variables indirectly depend upon terminal conditions and cannot be computed in analytically tractable closed form from the fundamental values. In such cases, the standard practice is to create a discrete lattice (often referred to as a state grid) for the evolution of the fundamental variables and then apply backward induction techniques to determine the values of the derived variables at the finite lattice points. However, MC methods with the sample paths drawn from a finite state grid are susceptible to parameter instability mentioned above. With a non-continuous mapping of random number draws to grid points, slight parameter changes will often cause discrete changes in a subset of the sample paths. This discontinuity may well result in disproportionate changes in the calculated average value.
The MC estimation process associated with a state grid will converge to the mathematical “true” value as the number of sample paths is increased and the grid is refined. However, the error in generic MC estimates converge to zero at a slow rate, with the expected error proportional to the inverse of the square root of the number of samples. For instance, a fourfold increase in the number of random samples is needed to reduce the expected error by one-half. With enhancements such as stratified sampling, control variates, and other variance reduction techniques, the error can be reduced but not rendered insignificant.
Accordingly, a need exists for an improved method and system of calculating derived variables and of determining actual and prospective changes in the value of securities, that are stable in the face of parametric changes